# birthday paradox

The birthday paradox is the fact – not really a paradox – that you need a group of only 23 people for there to be a better
than 50:50 chance that two of these people will have the same birthday.
This seems surprising because we are used to comparing* our particular
birthdays* with others and only rarely finding a perfect match. The
probability of any two individuals having the same birthday is just 1/365.
Even if you were to ask 20 people, the probability of finding someone with
your birthday is still less than 1/20. But the odds improve dramatically
when a group of people *ask each other* about their birthdays because
then there are many more opportunities for a match-up. One way to calculate
the probability of a birthday match is to count the pairs of people involved.
In a room of 23 people, there are (23 × 22)/2, or 253, possible pairs.
Each pair has a probability of success of 1/365 = 0.00274 (0.274%), and
thus a probability of failure of (1 - 0.00274) = 0.99726 (99.726%). The
probability of no match among any of the pairs of people is 0.99726 to the
253th power, which is 0.499 (49.9%). So the probability of a successful
match is (1 - 0.499), or slightly better than evens. With 42 people, the
probability of a birthday match climbs to 90%.